In this paper, we present a hybrid mathematical model describing crowd dynamics. More specifically, our approach is based on the well-established Helbing-like discrete model, where each pedestrian is individually represented as a dimensionless point and set to move in order to reach a target destination, with deviations deriving from both physical and social forces. In particular, physical forces account for interpersonal collisions, whereas social components include the individual desire to remain sufficiently far from other walkers (the so-called territorial effect). In this respect, the repulsive behaviour of pedestrians is here set to be different from traditional Helbing-like methods, as it is assumed to be largely determined by how they perceive the presence and the position of neighbouring individuals, i.e. either objectively as pointwise/localized entities or subjectively as spatially distributed masses. The resulting modelling environment is then applied to specific scenarios, that first reproduce a real-world experiment, specifically designed to derive our model hypothesis. Sets of numerical realizations are also run to analyse in more details the pedestrian paths resulting from different types of perception of small groups of static individuals. Finally, analytical investigations formalize and validate from a mathematical point of view selected simulation outcomes.

RSOS160561F5: Plot summarizing the simulation results. In the case of absence of field individuals (i.e. setting S1), the virtual pedestrian equally opts for each of the two lanes, in agreement with the corresponding experimental outcomes. When the groups of individuals strolling around are added within the areas around the flowerbed, the behaviour of the sample pedestrian is instead significantly affected by his/her type of perception. If he/she objectively considers all field individuals equally and objectively, i.e. as pointwise entities, he/she opts to use the less crowded lane on the left, as in the case of experimental setting S2(a). Finally, the empirical observations relative to the experimental settings S2(b,c) cannot be computationally reproduced with the use of a localized perception but with the use of a subjective distributed perception defined by the function w given in equation (3.19), i.e. which models the fact that the walker psychologically overestimates the presence of the field individuals.

Mentions:
To match the corresponding real-world configuration, 150 numerical realizations are then run for each simulation setting. As reproduced in figure 5, in the absence of any other individual (i.e. the repulsive velocity contribution is ), the simulated test walker 1 has the same probability to take either the left or the right lane. This is consistent with the dynamics of the corresponding experimental setting S1 (cf. the plot in figure 1c).Figure 5.

RSOS160561F5: Plot summarizing the simulation results. In the case of absence of field individuals (i.e. setting S1), the virtual pedestrian equally opts for each of the two lanes, in agreement with the corresponding experimental outcomes. When the groups of individuals strolling around are added within the areas around the flowerbed, the behaviour of the sample pedestrian is instead significantly affected by his/her type of perception. If he/she objectively considers all field individuals equally and objectively, i.e. as pointwise entities, he/she opts to use the less crowded lane on the left, as in the case of experimental setting S2(a). Finally, the empirical observations relative to the experimental settings S2(b,c) cannot be computationally reproduced with the use of a localized perception but with the use of a subjective distributed perception defined by the function w given in equation (3.19), i.e. which models the fact that the walker psychologically overestimates the presence of the field individuals.

Mentions:
To match the corresponding real-world configuration, 150 numerical realizations are then run for each simulation setting. As reproduced in figure 5, in the absence of any other individual (i.e. the repulsive velocity contribution is ), the simulated test walker 1 has the same probability to take either the left or the right lane. This is consistent with the dynamics of the corresponding experimental setting S1 (cf. the plot in figure 1c).Figure 5.

In this paper, we present a hybrid mathematical model describing crowd dynamics. More specifically, our approach is based on the well-established Helbing-like discrete model, where each pedestrian is individually represented as a dimensionless point and set to move in order to reach a target destination, with deviations deriving from both physical and social forces. In particular, physical forces account for interpersonal collisions, whereas social components include the individual desire to remain sufficiently far from other walkers (the so-called territorial effect). In this respect, the repulsive behaviour of pedestrians is here set to be different from traditional Helbing-like methods, as it is assumed to be largely determined by how they perceive the presence and the position of neighbouring individuals, i.e. either objectively as pointwise/localized entities or subjectively as spatially distributed masses. The resulting modelling environment is then applied to specific scenarios, that first reproduce a real-world experiment, specifically designed to derive our model hypothesis. Sets of numerical realizations are also run to analyse in more details the pedestrian paths resulting from different types of perception of small groups of static individuals. Finally, analytical investigations formalize and validate from a mathematical point of view selected simulation outcomes.